I am trying to solve
An agent learning to categorise news articles in two topics, World (W) and Finance (F). Out of $100$ articles, $40$ were classified as W, and $20$ of the articles were classified as both W and F. What is the value of conditional probability of an article being classified as F?*
The algorithm which I am assuming right for this question is
$$ P(A|B) = \frac{P(A\cap B)}{ P(B)}. $$
But how to specifically categorize F ? and take specific values for equation ? or I am assuming wrong formula ?
If the 40 W includes the number 20 W and F, then $$P(F|W) = \frac{P(FW)}{P(W)} = \frac{20/100}{40/100} = \frac{20}{40} = \frac{1}{2}.$$ If 40 W does not include the 20 W and F, then notice that $$P(W) = P(W\bar F)+P(WF) = \frac{40}{100}+\frac{20}{100} = \frac{60}{100}.$$ So, $$P(F|W) = \frac{P(FW)}{P(W)}= \frac{20/100}{60/100} = \frac{20}{60} = \frac{1}{3}.$$